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Mott insulators

Electronic Orbital Currents and Polarization in Mott Insulators; are electrons really localized? L.N . Bulaevskii , C.D. Batista , LANL M. Mostovoy, Groningen University D. Khomskii , Cologne University.

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Mott insulators

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  1. ElectronicOrbitalCurrentsandPolarizationinMottInsulators; are electrons really localized?L.N. Bulaevskii, C.D. Batista, LANLM. Mostovoy, Groningen UniversityD. Khomskii, Cologne University Introduction: why electrical properties of Mott insulators differ from those of band insulators. Magnetic states in the Hubbard model. Orbitalcurrents. Electronic polarization. Lowfrequencydynamic properties. Conclusions.

  2. Mott insulators Standard paradigm: for U>>t and one electron per site electrons are localized on sites. All charge degrees of freedom are frozen out; only spin degrees of freedom remain in the ground and lowest excited states

  3. Not the full truth! For certain spin configurations there exist in the ground state of strong Mott insulators spontaneous electric currents (and corresponding orbital moments)! For some other spin textures there may exist a spontaneous charge redistribution, so that <ni> is not 1! This, in particular, can lead to the appearance of a spontaneous electric polarization (a purely electronic mechanism of multiferroic behaviour) These phenomena, in particular, appear in frustrated systems, with scalar chirality playing important role

  4. Spin systems: often complicated spin structures, especially in frustrated systems – e.g. those containing triangles as building blocks ? Isolated triangles (trinuclear clusters) - e.g. in some magnetic molecules (V15, …) Solids with isolated triangles (La4Cu3MoO12) Triangular lattices Kagome Pyrochlore

  5. Triangular lattices: NaxCoO2, LiVO2,CuFeO2, LiNiO2, NiGa2S4, …

  6. Kagome:

  7. Spinels: The B-site pyrochlore lattice: geometrically frustrated for AF

  8. Often complicated ground states; sometimes spin liquids Some structures, besides, are characterized by: Vector chirality Scalar chirality - solid angle 1 1 - + 2 2 3 3  may be + or - :

  9. Scalar chiralityis often invoked in different situations: Anyon superconductivity Berry-phase mechanism of anomalous Hall effect New universality classes of spin-liquids Chiral spin glasses Chirality in frustrated systems: Kagome a) Uniform chirality (q=0) b) Staggered chirality (3x3)

  10. But what is the scalar chirality physically? What does it couple to? How to measure it? Breaks time-reversal-invariance T and inversion P - like currents! means spontaneous circular electric current and orbital moment 1 1 + - 3 2 2 3 Couples to magnetic field:

  11. DifferencebetweenMottandband insulators Only inthelimit electrons are localized on sites. At electrons can hop between sites. 1 2 Orbital current

  12. Spin current operator and scalar spin chirality Current operator for Hubbard Hamiltonian on bond ij: Projected current operator: odd # of spin operators, scalar in spin space. For smallest loop, triangle, Current via bond 23 On bipartite nn lattice is absent. 2 1 3 2 4 1 3

  13. Orbitalcurrentsinthespinorderedgroundstate Necessaryconditionfororbitalcurrentsisnonzeroaveragechirality Itmaybeinherenttospinorderingorinducedbymagnetic field . Triangleswith chirality Ontetrahedronchirality may be nonzero but orbital currents absent.

  14. Chiralityinthegroundstatewithoutmagneticordering Geometrically frustrated 2dsystem Mermin-Wigner theorem State with maximum entropy may be with broken discrete symmetry Example: model on kagome lattice: (group of C.Lhuillier)

  15. Ordering in model on kagome lattice at T=0 MonteCarloresults by Domenge et al., 2007. Phasediagram for classical order parameter b) Q=0 Neel Antiferro a) Cuboc Antiferro Ferro Neel Antiferro b) For S=1/2 for low T cub'oc is chiral with orbital currents and without spin ordering.

  16. Boundary and persistent current x 2 4 6 y 4 5 6 z 1 2 3 1 5 3 Boundarycurrent in gaped 2d insulator

  17. Tetrahedra in exact solution: Ground state - S=0, doubly-degenerate. In the ground state one can chose the state with chirality + or - . Nonzero chirality magnetic state. But currents at each edge = 0 ! magnetic octupole states? Very similar to the situation with doubly-degenerate eg orbitals: |z2> ------- Tz=1/2 |x2-y2> --- Tz=-1/2 (|z2>+i |x2-y2>) ---- Ty =1/2, (|z2>-i |x2-y2>) ---- Ty =-1/2, Eigenstates of Ty– states withmagnetic octupoles! Real combinations a|z2>+b |x2-y2> – states withelectric quadrupoles. The same for spin tetrahedra ?

  18. Spin-dependent electronicpolarization Chargeoperatoronsitei: Projected charge operator Polarization on triangle Charge onsiteiissumovertrianglesatsitei. 2 1 3

  19. Electronicpolarization on triangle or 1 P 3 singlet 2 Purely electronic mechanism of multiferroic behavior!

  20. Charges on kagome lattice Current ordering induced by perp. magnetic field 1/2 magnetization plateau: Chargeordering for spins 1/2 in magnetic field: spin-driven CDW Typical situation at the magnetization plateaux!

  21. Diamond chain (azurite Cu3(CO3)2(OH)2 ) spin singlet -will develop S-CDW Saw-tooth (or delta-) chain Net polarization Net polarization

  22. Isolated triangle: accounting for DM interaction DMcoupling: ForV15 Splitslowestquartetinto2doubletsandseparatedbyenergy Acelectricfieldinducestransitionsbetween Acmagneticfieldinducestransitionsbetween

  23. ESR : magnetic field (-HM) causes transitions Here: electric field (-Ed) has nondiagonal matrix elements in : electric field will cause dipole-active transitions -- ESR caused by electric field E ! E and H E H E H

  24. Lowfrequencydynamicproperties: negativerefractionindex Responsestoacelectricandacmagneticfieldare comparablefor100K: Spin-orbitalcouplingmayleadtocommonpolesin and Negativerefractionindexifdissipationisweak.

  25. CONCLUSIONS Contrary to the common belief, there are real charge effects in strong Mott insulators (with frustrated lattices): spin-driven spontaneous electric currents and orbital moments, and charge redistributionin the ground state Spontaneous currents are ~ scalar spin chirality Charge redistribution ( <ni> is not 1!) may lead to electric polarization ( purely electronic mechanism of multiferroicity) Many consequences: In the ground state: lifting of degeneracy; formation of spin-driven CDW, ……. In dynamics: electric field-induced "ESR"; rotation of electric polarization by spins; contribution of spins to low-frequency dielectric function; possibility of negative refraction index; etc

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